Criteria for the validity of Amontons Coulomb’s law;

Study of friction　

using dynamics of driven vortices of superconductor Department of Basic

Sciences, University of Tokyo

Atsutaka MAEDA Daisuke Nakamura

Ryo Tanaka Y. Imai

ISSP International Workshops on Soft Matter Physics –structural rheology- (2010.8.10, ISSP, Chiba, JAPAN)

JAERI S. Okayasu

RIKEN

S. Savelev F. Nori

CRIEPI I. Tsukada

Ecole Polytechnique M. Konczykowski C. J. van der Beek

　　　　　　　　　　　　　　　　　 　Outline 1)　Background： Problems in physics of friction Collective dynamics of quantum condensate 　　　　Dynamics of driven vortices in superconductors

2)　Physics of friction by using dynamics of driven vortices 　　　　Model 　　　　Kinetic friction as a function of velocity 　　　　　　　　Broadened dynamical phase transition

　　　　Static friction as a function of waiting time 　　　　　　　　Criteria for the validity of Amontons-Coulomb friction

3)  Response to ac (µ-wave) driving force

4)　Conclusion, perspective

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

D. Nakamura et al.: arXiv 0906.3086.

microscopic friction

macroscopic friction

Microscopic friction vs Macroscopic friction

D. M. Tanner et al., MEMS Reliability : Infrastructure, Test Structures, Experiments, and Failure Modes 2000-2091 (Sandia National Laboratories, 2000).

Amontons-Coulomb’s law (A-C law)

What is the limit of the validity of Amontons-Coulomb’s law?

Fs is constant Fk does not depend on v

Fs : maximum static friction force Fk : kinetic friction force

In some cases, even the macroscopic friction

does not obey the A-C law

Amontons-Coulomb friction vs real friction Real friction system Amontons-Coulomb’s law

To understand friction from microscopic point of view　　　

Need model system with which systematic experiments are possible in a repeated manner (free from wear)

1 D model for clean surfaces

clean surface (normal) dirty surface

・clean surface finite Fk even for zero Fs ・disordered surface less velocity dependent 　　similar to Amontons-Coulomb’s law

numerical solution for the above equation Fk as a function of velocity

Microscopic formulation of friction

steady state summing up for all atoms time averaged friction: sum of interatomic (pinning) forces

eq. motion for a lower atom i

　　　　　　　　　　　　　　　　　←a　displacement of upper atom i　ui , mass　ma 　　　　　　　　　　　　　　　　 ←b　 displacement of lower atom i　vi , mass　mb

eq. motion for an upper atom i

dissipation from a representative DF to others

H. Matsukawa and H. Fukuyama: PRB 49, 17286 (1994)

Model systems for friction study in quantum condensate in solids

Charge-density wave (CDW)

Vortex lattice of superconductor

ui : displacement of i-th electron in the CDW m: mass of the i-th electron Fp: pinning force for i-th electron　　　　　　　　　　

ui : displacement of i-th votex in the lattice m: mass of the i-th vortex in the lattice Fp: pinning force for i-th votex　　　　　　　　　　

B

1D

2D

flux flow

creep motion

Vortex dynamics - the ideal model system of friction

A. Maeda et al., PRL 94, 077001 (2005). Advantage ・ scan H, T and F continuously ・ intrinsically reproducible --- no wear, no debris

H. Matsukawa and H. Fukuyama PRB 49, 17286 (1994).

・ multi-internal-degrees of freedom ・ non-equilibrium ・ non-linearity

Common feature

flux flow elastic pinning

driving force　J ×Φ0

viscous force η<ｖ>

moving direction

pinning force FPIN

J, E

creep motion Friction force on solid-solid interface

Driven vortices as a model system of solid-solid friction

Purpose of study

(1) Measure kinetic friction as a function of velocity

(2) Static friction as a function of waiting time

(3) Find the criteria for the validity of Amontons-Coulomb friction

High-Tc cuprate superconductor：variety of H-T phase diagram

Change temperature, magnetic field, pinning, system size

(4) Ac dynamics (µ-wave to THz)

Microscopic understanding of elementary excitation at the interface

Expressing solid-solid friction in terms of vortex motion

Driving force 　J ×Φ0

viscous force η<ｖ>

direction of vortex motion

kinetic friction FFRIC （pinning force）

I -V measurement and viscosity ,η , measurement can deduce kinetic friction

Flux flow resistivity

necessary to make correspondence with theory

J: current density ρ: resistivity

Φ0: flux quantum

La2-xSrxCuO4 （LSCO） (x = 0.12, 0.15)

Sample preparation

Crystal structure thickness　～ 3000 Å

single crystal, thin film

PLD method

photolithography ＋ chemical etching

5.8 GeV Pb ion for the columnar defects @ Grand Accelerateur National d’Ions Lourds(GANIL)

Ecole Polytechnique, M. Konczykowski and C. J. van der Beek

S. Komiyama (Univ. of Tokyo）

Pb ion

Rectangular

Bridge

Bridge wih columnar defects

Fabrication of samples ～ 0.5mm

～ 1.5mm

Fk (v) （up to ～1 km/s）

4) smaller Fk in irradiated samples

3) Fk saturates and decreases

inconsistent with the behavior at low velocities ?

pristine 3T irradiated

2) very much different from the Amontons-Coulomb behavior

1) Fk changes with B and T in a reproducible manner

good as a model system

similar to “clean surface”

existence of a peak in Fk(v)

Data points with crosses denote pulsed measurements

200 MeV Iodine

BΦ=3T Columnar defects

La2-xSrxCuO4 A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Bi2Sr2CaCu2Oy (bulk single crystals)

Universal to many SCs

Minimal model to explain the data : overdamped equation of motion

: position of vortices

: viscosity of vortices

: substrate pinning potential

: inter-vortex interaction

: driving force

: thermal random force

: temperature

S. Savel’ev and F. Nori

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Numerical simulation for 1D vortex array at finite temperatures S. Savel’ev and F. Nori

a peak

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

A peak in the kinetic friction Fk(v)

velocity at the peak S. Savel’ev and F. Nori

in good agreement with experiment

Potential energy plays an important role for Fk(v).

“Inversion” of kinetic friction at intermediate velocities！

sample with strong pinning higher static friction lower kinetic friction 　more gradual dependence on v

velocity

friction

3Ｔirradiated

pristine

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

S. Savel’ev and F. Nori

Q/l～ηv ~200m/s

Physical origin of the peak

changing parameters change transition between static and kinetic regime

increasing magnetic field increasing temperature decreasing system size (macro to micro)

broaden the transition

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

v

F k

static

kinetic

N strongly coupled system

collective coordinate

new stochastic variable

effective temperature

(L : system size)

Suggesting the importance of thermal fluctuation in microscopic friction

Time domain response for various tw

Input sawtooth signal, changing t W , measure Ic

µ

D. Nakamura et al.: arXiv 0906.3086.

Driving current

Measurement of Fs(tw)

Waveforms become different with tW

Ic depends on T → normalize by Ic(max tW)

Rectangular-type thin film

Low T

～10-1 s

Logarithmic Characteristic timescale

x Tc (K) # rectangular A 0.12 35.20 # rectangular B 0.15 35.06

characteristic timescale

log tw

Tg

High T

D. Nakamura et al.: arXiv 0906.3086.

The origin of logarithmic dependence near Tg

Thermally Assisted Flux Flow P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39(1964). P. H. Kes et al., Supercond. Sci. Technol. 1, 242(1989).

In TAFF region, spatial magnetic profile obeys diffusion equation of motion

log tw at high T

Blatter et al. RMP (94).

This timescale can be estimated as

At I < Ic, Critical state

D. Nakamura et al.: arXiv 0906.3086.

depinning energy

～10-1 s

Rapid relaxation at low T with characteristic timescale

Bundle relaxation

energy loss by the viscous motion

h : thickness, η: viscous coefficient ξ: coherence length

Length scale of coherent relaxation (cf. vortex-vortex pacing : ~ 50nm)

Dynamic coherence length : 30µm (from noise study in Bi-2212)

A. Maeda et al., Phys. Rev. B 65, 054506 (2002).

# rect. A

Low temp.

Bundle relaxation picture confirmed

Slow tW* cannot be explanined by the motion of a single vortex

D. Nakamura et al.: arXiv 0906.3086.

Interplay between the size of coherent bundle and the system size crucially affects the validity of Amontons-Coulomb’s law

Size effect of the relaxation

no relaxation by the edge pinning effect

w

smaller relaxation region for the narrower bridge sample

crossover line of the relaxation µ

D. Nakamura et al.: arXiv 0906.3086.

Bridge sample with columnar defects

Sample with columnar defects → no relaxation at all

Strong pinning force also leads to the Amontons-Coulomb’s type friction

x Tc (K) # Columnar 0.15 35.11 2.0

Dramatic changing for the relaxation

D. Nakamura et al.: arXiv 0906.3086.

high T low T

Rectangular film log tW characteristic timescale

Bridge film log tW no relaxation

Bridge film with columnar defects no relaxation no relaxation

Summary of Fs(tw) experiments The size of vortex bundle

size effect

columnar pin effect

Thermally Assisted Flux Flow of vortex bundle Bundle size determines the characteristic timescale no relaxation very strong pinning center

The origin of the relaxation phenomena

D. Nakamura et al.: arXiv 0906.3086.

(Amontons-Coulomb like)

Our achievements on this model approach

Key parameters of the validity of Amontons-Coulomb’s law ・ pinning strength and thermal fluctuation ・ size of the coherently moving object and system size

Bundle size

Inte

ract

ion

(pin

ning

)

Degrees of freedom

Amontons-Coulomb’s type friction

a universal parameter :

a criteria of the validity of A-C law :

Fs(tW) Fk(v)

D. Nakamura et al.: arXiv 0906.3086.

　　Ac dynamics at the interface vs microscopic understanding of friction

1)　Importance of lattice dynamics 　　　 Phonons 　　　　Phonons in disordered systems 　　　　Corresponding excitations in CDWs and vortices in SC

2) data at µ-waves for vortices

Friction at the interface Dissipation by elementary excitation at the interface

Friction vs ac dynamics

Normal modes by periodic array of atoms: Phonon

(C. Kittel : “Introduction to Solid State Physics”)

Dispersion relation Angular frequency Wave number (well defined)

Optical response by a bound mode: Lorentz oscillator

(F. Wooten : “Optical Properties of Solids”)

for small ω

For pinned CDW H. Fukuyama and P. A. Lee : Phys. Rev. B17 (1978) 535. P. A. Lee, T. M. Rice and P. W. Anderson: SSC 14 (1974) 703.

(D. Reagor et al.: Phys. Rev. B34 (1986) 2212.) (W. Wu et al.: Phys. Rev. Lett. 52 (1984) 2382.)

(R. J. Cava et al.: Phys. Rev. B31 (1985) 8325.)

(also J. P. Stokes et al.: Phys. Rev. B32 (1985) 6939.)

(R. J. Cava et al.: Phys. Rev. B30 (1984) 3228.)

Experiments in CDWs

Mean-field model of vortex motion P. Martinoli et al. Physica B 165&166, 1163 (1990), M. W. Coffey and J. R. Clem: PRL 67 (1991) 386, PRB 45 (1992) 9872.

remarks

(a) Magnus force, vortex-vortex interaction are not considered. (b) Vortex mass is considered to be zero.

(d) From microscopic point of view, the two-fluid picture is not a good approximation. (Eschrig-Rainer-Sauls)

(c) Condensate wave function is considered to be s-wave.

driving force　J ×Φ0

viscous force η<ｖ>

moving direction

pinning force FPIN

J, E overdamped massless

Ac resistivity by the vortex motion

dissipative (flux flow)

reactive (pinned)

Our expectation

Non AC like

AC like

Stronger pinning Larger bundle size

Weaker pinning Smaller bundle size

Digergent ε(ω) Remarkable nonlinearity

Lorentz-oscillator like ε(ω) no nonlinearity

B. Parks et al. Phys. Rev. Lett. 74 (1995) 3265.

Experiments in vortices in superconductors (THz)

YBCO

overdamped

Well described by the mean-field model with field independent ωco

High-Tc superconductor (Y. Tsuchiya et al.: Phys. Rev. B63 (2001) 184517-1.) (Y. Matsuda et al.: PRB66(2002)014527.)

TBCO YBCO

Conventional superconductor

Experiments in vortices in superconductors (µ-wave)

Pb-In, Nb-Ta (J. C. Gittleman and B. Rosenblum: Phys. Rev. Lett. 16 (1968) 734.)

overdamped

Seems to exhibit no apparent nonlinearity at present

Need to be investigated systematically in more detail

Ac impedance of SC: vortex motion plus superfluid M. W. Coffey and J. R. Clem, PRL 67, 386 (1991).

δ nf : normal-fluid skin depth λ : penetration depth

flux flow

normal-fluid response

Parameters η : viscous drag coeff. τp

–1 : pinning frequency ε : flux-creep factor λ 　 : penetration depth

from ρ1

from ρ2

coax. cable

modified 2.4mm J-J adapter

Sample with Corbino disk geometry

Measurement of S11(ω) by Vector Network Analyzer

LSCO: viscous drag η

Dissipation: similar to other cuprates (YBCO, BSCCO etc.)

η depends on B.

A. Maeda et al.: JPSJ 76 (2007) 094708. Y. Tsuchiya et al.: PRB 63 (2001) 184517.

A. Maeda et al.: Physica C 460-562 (2007) 1201.

Conclusion

（１）Discuss dynamics of driven vortices and physics of friction utilizing driven vortices of superconductor

（２）Kinetic friction as a function of velocity: understood in terms of simple overdamped oscillator model Broadened transition of dynamic phase transition (Amontons-Coulomb behavior) suggesting the importance of thermal fluctuation for microscopic friction

（３）Static friction as a function of waiting time: relaxation of vortex bundle 　　　 stronger pinning, large bundle size ⇔　Amontons-Coulomb like behavior

（４）Criteria for the validity of Amontons-Coulomb friction

Presence of a universal parameter :

（５）Ac dynamics (µ-wave to THz) of vortices 　　　 no remarkable anomaly nor nonlinearity such as CDW system at present

Future perspective

（１）Change pinning, system size more systematically introduce stick-slip motion in driven vortices change velocity dependence of kinetic friction largely Feedback to physics of friction

‐Basic step for understanding friction, control of friction‐

（２）Need to investigate correspondence between ac dynamics and other friction properties 　　　　　　　　　　　‐comparative studies of Fk(v), Fs(tw) and σ(ω)‐